Properties

Modulus 35
Conductor 35
Order 12
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 35.l

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(35)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3,8]))
 
pari: [g,chi] = znchar(Mod(32,35))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 35
Conductor = 35
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 12
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 35.l
Orbit index = 12

Galois orbit

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

\(\chi_{35}(2,\cdot)\) \(\chi_{35}(18,\cdot)\) \(\chi_{35}(23,\cdot)\) \(\chi_{35}(32,\cdot)\)

Values on generators

\((22,31)\) → \((i,e\left(\frac{2}{3}\right))\)

Values

-1123468911121316
\(-1\)\(1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(1\)\(-i\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(-i\)\(e\left(\frac{1}{3}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{12})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 35 }(32,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{35}(32,\cdot)) = \sum_{r\in \Z/35\Z} \chi_{35}(32,r) e\left(\frac{2r}{35}\right) = -5.0218389077+3.1274804532i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 35 }(32,·),\chi_{ 35 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{35}(32,\cdot),\chi_{35}(1,\cdot)) = \sum_{r\in \Z/35\Z} \chi_{35}(32,r) \chi_{35}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 35 }(32,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{35}(32,·)) = \sum_{r \in \Z/35\Z} \chi_{35}(32,r) e\left(\frac{1 r + 2 r^{-1}}{35}\right) = 3.5256696496+0.9447003354i \)